Calculus Examples

Find the Area Between the Curves 2x+y^2=8 , x=y
,
Step 1
Solve by substitution to find the intersection between the curves.
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Step 1.1
Replace all occurrences of with in each equation.
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Step 1.1.1
Replace all occurrences of in with .
Step 1.1.2
Simplify the left side.
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Step 1.1.2.1
Multiply by .
Step 1.2
Solve for in .
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Step 1.2.1
Subtract from both sides of the equation.
Step 1.2.2
Factor the left side of the equation.
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Step 1.2.2.1
Let . Substitute for all occurrences of .
Step 1.2.2.2
Factor using the AC method.
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Step 1.2.2.2.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 1.2.2.2.2
Write the factored form using these integers.
Step 1.2.2.3
Replace all occurrences of with .
Step 1.2.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 1.2.4
Set equal to and solve for .
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Step 1.2.4.1
Set equal to .
Step 1.2.4.2
Add to both sides of the equation.
Step 1.2.5
Set equal to and solve for .
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Step 1.2.5.1
Set equal to .
Step 1.2.5.2
Subtract from both sides of the equation.
Step 1.2.6
The final solution is all the values that make true.
Step 1.3
Replace all occurrences of with in each equation.
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Step 1.3.1
Replace all occurrences of in with .
Step 1.3.2
Simplify the left side.
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Step 1.3.2.1
Remove parentheses.
Step 1.4
Replace all occurrences of with in each equation.
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Step 1.4.1
Replace all occurrences of in with .
Step 1.4.2
Simplify the left side.
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Step 1.4.2.1
Remove parentheses.
Step 1.5
The solution to the system is the complete set of ordered pairs that are valid solutions.
Step 2
Solve in terms of .
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Step 2.1
Subtract from both sides of the equation.
Step 2.2
Divide each term in by and simplify.
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Step 2.2.1
Divide each term in by .
Step 2.2.2
Simplify the left side.
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Step 2.2.2.1
Cancel the common factor of .
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Step 2.2.2.1.1
Cancel the common factor.
Step 2.2.2.1.2
Divide by .
Step 2.2.3
Simplify the right side.
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Step 2.2.3.1
Simplify each term.
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Step 2.2.3.1.1
Divide by .
Step 2.2.3.1.2
Move the negative in front of the fraction.
Step 3
The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.
Step 4
Integrate to find the area between and .
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Step 4.1
Combine the integrals into a single integral.
Step 4.2
Multiply by .
Step 4.3
Split the single integral into multiple integrals.
Step 4.4
Apply the constant rule.
Step 4.5
Since is constant with respect to , move out of the integral.
Step 4.6
Since is constant with respect to , move out of the integral.
Step 4.7
By the Power Rule, the integral of with respect to is .
Step 4.8
Since is constant with respect to , move out of the integral.
Step 4.9
By the Power Rule, the integral of with respect to is .
Step 4.10
Simplify the answer.
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Step 4.10.1
Combine and .
Step 4.10.2
Substitute and simplify.
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Step 4.10.2.1
Evaluate at and at .
Step 4.10.2.2
Evaluate at and at .
Step 4.10.2.3
Evaluate at and at .
Step 4.10.2.4
Simplify.
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Step 4.10.2.4.1
Multiply by .
Step 4.10.2.4.2
Multiply by .
Step 4.10.2.4.3
Add and .
Step 4.10.2.4.4
Raise to the power of .
Step 4.10.2.4.5
Combine and .
Step 4.10.2.4.6
Raise to the power of .
Step 4.10.2.4.7
Multiply by .
Step 4.10.2.4.8
Combine and .
Step 4.10.2.4.9
Combine the numerators over the common denominator.
Step 4.10.2.4.10
Add and .
Step 4.10.2.4.11
Cancel the common factor of and .
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Step 4.10.2.4.11.1
Factor out of .
Step 4.10.2.4.11.2
Cancel the common factors.
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Step 4.10.2.4.11.2.1
Factor out of .
Step 4.10.2.4.11.2.2
Cancel the common factor.
Step 4.10.2.4.11.2.3
Rewrite the expression.
Step 4.10.2.4.11.2.4
Divide by .
Step 4.10.2.4.12
Multiply by .
Step 4.10.2.4.13
Combine and .
Step 4.10.2.4.14
Cancel the common factor of and .
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Step 4.10.2.4.14.1
Factor out of .
Step 4.10.2.4.14.2
Cancel the common factors.
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Step 4.10.2.4.14.2.1
Factor out of .
Step 4.10.2.4.14.2.2
Cancel the common factor.
Step 4.10.2.4.14.2.3
Rewrite the expression.
Step 4.10.2.4.14.2.4
Divide by .
Step 4.10.2.4.15
Subtract from .
Step 4.10.2.4.16
Raise to the power of .
Step 4.10.2.4.17
Cancel the common factor of and .
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Step 4.10.2.4.17.1
Factor out of .
Step 4.10.2.4.17.2
Cancel the common factors.
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Step 4.10.2.4.17.2.1
Factor out of .
Step 4.10.2.4.17.2.2
Cancel the common factor.
Step 4.10.2.4.17.2.3
Rewrite the expression.
Step 4.10.2.4.17.2.4
Divide by .
Step 4.10.2.4.18
Raise to the power of .
Step 4.10.2.4.19
Cancel the common factor of and .
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Step 4.10.2.4.19.1
Factor out of .
Step 4.10.2.4.19.2
Cancel the common factors.
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Step 4.10.2.4.19.2.1
Factor out of .
Step 4.10.2.4.19.2.2
Cancel the common factor.
Step 4.10.2.4.19.2.3
Rewrite the expression.
Step 4.10.2.4.19.2.4
Divide by .
Step 4.10.2.4.20
Multiply by .
Step 4.10.2.4.21
Subtract from .
Step 4.10.2.4.22
Multiply by .
Step 4.10.2.4.23
Add and .
Step 5